Evaluate the integrals in Exercises 77–90.
84. ∫(from 2 ...

Question

Evaluate the integrals in Exercises 77–90.

84. ∫(from 2 to 4)2dx/(x²-6x+10)

Accepted Answer

Welcome back everyone. Evaluate integral from 123 of 4 d x divided by x 2 minus 4 x plus 8. For this problem we are going to make use of the integral duu divided by u2 plus a 2. Remember the value of this integral. It is 1 divided by a inverse of tangent of u divided by a plus a constant of integration c. In order to achieve this integral, we're going to complete the square. So let's analyze the quadratic polynomial x2 minus 4 x + 8, and let's complete the square. That'll be x minus 22 plus 4. So we can now rewrite the integral as integral from 1 up to 3 of. We can factor out 4 because it's a constant and simply write d x divided by x minus 22 + 4. Now let's introduce a u substitution and let's suppose that u is equal to x minus 2. The next step is to find the derivative of u with respect to x. That's the derivative of x minus 2, which is 1, which essentially means that duu is equal to dx, right? And now let's change the limits of integration. So the lower one is going to be 1 minus 2, which is -1. And the upper limit of integration u2 is going to be 3 minus 2, which is 1. So we can now rewrite the integrals for. Integral from -1 up to 1, Dx becomes DU. And in the denominator we have u2 + 4. We can also rewrite 4 as 22 so that we clearly understand that in this case a is 2. And now we can apply the formula, so that would be or. Our integral is going to be equal to 1 divided by a, which is 1 divided by 2. Inverse of tangent of u divided by 2. And we are going to evaluate the result from -1 up to 1. Notice that 4 divided by 2 gives us 2, so we can basically write 2 inverse of tangent of you divided by 2 from -1 up to 1. And now let's calculate the results. That's 2. Inverse sub tangent of 1/2 minus 2. Inverse of tangent of -1/2. And now we know that inverse of tangent is an odd function, so we can basically rewrite it as 2. Inverse of tangent of 1/2. Minus -2. Inverse of tangent of positive 1/2. Combining we get 4 inverse tangents. Of one half, which is our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 77–90.84. ∫(from 2 to 4)2dx/(x²-6x+10)

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Evaluate the integrals in Exercises 77–90.
84. ∫(from 2 to 4)2dx/(x²-6x+10)